Tagged Questions

Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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2
votes
2answers
397 views

What does [n] mean here?

I am reading this document. What is the meaning of $[n]$ ? Is it power set of $\{1,2,3...n\}$?
5
votes
2answers
1k views

Proving a fourier series identity

I studied fourier series as an undergrad and grad. student in EE but did not fully grasp the concepts. Now that I am involved in medical imaging (MRI) understanding the basics of fourier series and ...
4
votes
1answer
317 views

Convergence of $\lim \limits_{n,v \rightarrow \infty} \int_0^1 F_n (x) e^{-i2\pi v x} \mbox{d} x $

This is a stronger one related to the question Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x $. $F_n(x) : [0,1] \rightarrow \bf R $, for $1 \leq i \leq ...
3
votes
1answer
168 views

Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x $

$f_n(x) : [0,1] \rightarrow \bf R $, and $f_n$ is $\frac{1}{n}$-periodic, $\max f_n(x) = n$, $\min f_n(x) = -n$. As $n$ and $v$ goes to infinity simultaneously, prove the convergence of ...
6
votes
2answers
438 views

Equidistributed sequence and Riemann integrable function

Let $f$ be a function of period 1, Riemann integrable on [0,1]. Let $\xi_n$ be a sequence which is equidistributed in $[0,1)$. (a) Is it true that $$\frac{1}{N}\sum_{n=1}^N f(x+\xi_n)$$ converges to ...
4
votes
2answers
230 views

Is this a bounded sequence?

Let $0<a\le 1$ be fixed. Is the sequence $$a_N=\int_1^N x^{-a}e^{ix} dx$$ bounded?
14
votes
3answers
904 views

learning algebra and harmonic analysis

I've revised my question a bit in response to the (very helpful) advice so far-- I have an engineering background but am interested in learning abstract harmonic analysis. My interest is rather ...
5
votes
1answer
557 views

Isoperimetric inequality implies Wirtinger's inequality

Let $C: x=x(t), y=y(t), a\le t\le b$ be a $C^1$ closed curve (not necessarily simple).The isoperimetric inequality says that $$ A\le \frac{\ell^2}{4\pi},$$ where $$A=\left|\int_C y(t)x'(t) ...
4
votes
2answers
773 views

Can you provide help with interpreting this periodogram?

I'm trying to track down the source of some wonky data. The data are response times (RTs) collected from humans using a computer keyboard. Here's a histogram of the RTs, binned to 1ms: Obvious is a ...
3
votes
2answers
417 views

FFT gives ghost frequency

Imagine the varying frequency generator, which feeds this frequency to the oscilloscope. Oscilloscope is turned to show signal spectra with FFT. The image on the screen of the oscilloscope is as ...
3
votes
2answers
2k views

Fourier cosine transform

Find Fourier cosine transform of $e^{-a^2 x^2}$ and hense evaluate Fourier sine transform of $x\cdot e^{-a^2x^2}$. I can solve this question only if there is $x$ instead of $x^2$ in the exponential ...
1
vote
3answers
469 views

Fourier analysis for waves

I'm studying physics, so I'm sorry if I'll write some inexact things in this post. I wish you can understand me. If we have 1D wave equation: $$\frac{\partial^2 \psi}{\partial ...
3
votes
2answers
406 views

A sequence not equidistributed in [0,1]

Let $$a_n=\left(\frac{1+\sqrt{5}}{2}\right)^n.$$ For a real number $r$, denote by $\langle r\rangle$ the fractional part of $r$. Why is the sequence $$\langle a_n\rangle$$ not equidistributed in ...
3
votes
1answer
123 views

A limit related to the Gibbs phenomenon

Let $$D_N(x)=\frac{\sin [(N+(1/2))t]}{\sin (t/2)}$$ be the Dirichlet kernel. Let $x(N)$ be the number in $0<x<\pi/N$ such that $D_N(x)=1$. Is $$\left|\int_{x(N)}^{\pi/N} D_N(t)\mathrm dt ...
2
votes
1answer
144 views

Getting the value of a Fourier Transform, problem with the complex part

I'm currently trying to do some Fourier transformations, or at least trying to understand them. The only thing I'm worried about is the complex part of the function. All I have is some basic, self ...
4
votes
4answers
559 views

Fourier series analog of a formula in Fourier transform

Every Fourier transform formula that I know of has a corresponding Fourier series analog, except the multiplication formula $$\int_{-\infty}^\infty f(x)\hat{g}(x) dx=\int_{-\infty}^\infty ...
8
votes
2answers
11k views

What is the Fourier transform of the product of two functions?

Given $x(t) = f(t) \cdot g(t)$, what is the Fourier transform of $x(t)$? If possible, please explain your answer. The motivation behind the question is homework, but this is a basic principle in ...
18
votes
1answer
823 views

Accessible proof of Carleson's $L^2$ theorem

Lennart Carleson proved Luzin's conjecture that the Fourier series of each $f\in L^2(0,2\pi)$ converges almost everywhere. Also, Richard Hunt extended the result to $L^p$ ($p>1$). Some time ago I ...
15
votes
5answers
876 views

Interpretation of Poisson Summation Formula

This question arises from a Fourier transform class I took about a year back. The poisson summation formula is: $$\displaystyle \sum_{n= - \infty}^{\infty} f(n) = \displaystyle \sum_{k= - ...
2
votes
1answer
1k views

how to calculate fourier transform of a power of radial function

Is there an easy way to see that the $n$-dimensional Fourier transform of $1/|x|^a$ is equal to $1/|x|^{n-a}$ (up to multiplicative constant) where $x$ is an $n$ dimensional vector (assuming that $ 0 ...
6
votes
4answers
5k views

Recommended books/links for Fourier Transform beginners?

I am a student taking engineering course and wish to learn more about Fourier Transforms. It seems very useful. Would highly appreciate it if anyone could advise me where to start.
6
votes
3answers
767 views

Is $L^2(\mathbb{R})$ with convolution a Banach Algebra?

Is $L^2(\mathbb{R})$ a Banach algebra, with convolution? I am pretty sure the answer is no, because I think that $f,g \in L^2(\mathbb{R})$ does not imply that $f*g \in L^2(\mathbb{R})$. However, I ...
5
votes
1answer
2k views

Does rapid decay of Fourier coefficients imply smoothness?

Under the isomorphism of Hilbert spaces $L^2(S^1)\to\ell^2(\mathbb Z),\quad e^{2\pi i n t}\mapsto e_n$, smooth functions on the circle are mapped to rapidly decaying sequences (see wikipedia). Is the ...
6
votes
1answer
655 views

Pointwise but not uniform convergence of a Fourier series

What is an example of a continuous, or even better, differentiable, $2\pi$ (or 1) periodic function whose Fourier series converges pointwise but not uniformly? (Such function cannot be of Hölder ...
5
votes
4answers
837 views

Convergence of a Fourier series

Let $f$ be the $2\pi$ periodic function which is the even extension of $$x^{1/n}, 0 \le x \le \pi,$$ where $n \ge 2$. I am looking for a general theorem that implies that the Fourier series of $f$ ...
8
votes
1answer
822 views

How do I compute the eigenfunctions of the Fourier Transform?

In Andy's answer to the question "What are fixed points of the Fourier Transform" on Math Overflow, he shows that the Fourier Transform has eigenvalues $\{+1, +i, -1, -i \}$ and that the projections ...
1
vote
3answers
3k views

How to sketch a sinc function by hand?

I have to do this for an upcoming exam, but cannot find anywhere (in the textbook or online) how to do this. I only really need to know a couple points to plot it... when x = 0, and then the earliest ...
4
votes
3answers
489 views

Derivatives distribution

Let $f$ be a distribution on $\mathbf{R}^n$ (in the Schwartz sense) such that $$\frac{\partial f}{\partial x_i} = 0 \text{ for $i = 1, \ldots, n$.}$$ Then how to prove that $f$ is a constant? I had ...
3
votes
1answer
370 views

Fourier transform of a special Schwartz function

In Classical Fourier Analysis by Loukas Grafakos we have in Proposition 2.3.25 the following definition for $\mathcal{S}_\infty(\mathbf{R}^n)$, namely that these are all the Schwartz functions $\phi$ ...
10
votes
4answers
387 views

Can the phase of a function be extracted from only its absolute value and its Fourier transform's absolute value?

If for a function $f(x)$ only its absolute value $|f(x)|$ and the absolute value $|\tilde f(k)|$ of its Fourier transform $\tilde f(k)=N\int f(x)e^{-ikx} dx$ is known, can $f(x) = |f(x)|e^{i\phi(x)}$ ...
2
votes
1answer
663 views

Rigorous definition of convolution with the unit doublet

The unit doublet is a symbolic object whose convolution with a differentiable function is supposed to give the derivative: $$(x * u_1)(t) = \frac{dx(t)}{dt}$$ See also: ...
10
votes
5answers
3k views

Extracting exact frequencies from FFT output

Say I pass 512 samples into my FFT My microphone spits out data at 10KHz, so this represents 1/20s. (So the lowest frequency FFT would pick up would be 40Hz). The FFT will return an array of 512 ...
238
votes
22answers
21k views

Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

As I have heard people did not trust Euler when he first discovered the formula $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs. I ...
5
votes
1answer
337 views

Let $\psi$ be a wavelet. Can its Fourier transform $\hat{\psi}$ be also wavelet?

Let $\psi$ be a wavelet. Can its Fourier transform $\hat{\psi}$ be also wavelet? produce an example or prove that it is not possible. A wavelet is a function $\psi:\mathbb R\to\mathbb R$ such that (i) ...
-2
votes
1answer
347 views

show that $\int_{-\infty}^{\infty}\frac{(\sin(t))^3}{t^3}dt=\frac{3\pi}{4}$

Use fourier transform and convolution to show that $$\int_{-\infty}^{\infty}\frac{(\sin(t))^3}{t^3}dt=\frac{3\pi}{4}$$ convolution is defined by $$(f * g)(t) = \int_{-\infty}^{\infty}(f(s)*g(t-s)ds ...
0
votes
2answers
2k views

Compute the Fourier transform of $f(t) = \sin t$?

Compute the Fourier transform of $f(t) = \sin t$. Does that converge?
10
votes
1answer
214 views

Ergodic flow in tori

Let $\mathbb{T}^n = { (z_1,\ldots,z_n) \in \mathbb{C}^n : |z_l| = 1, \; 1 \leq l \leq n }$ denote the $n$-torus, and let $t_1, \ldots, t_n$ be arbitrary real numbers. Then it can be shown that the ...
46
votes
4answers
7k views

Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. My question: What is the connection between these two? Is there a way to get from one to the other (and ...
5
votes
1answer
444 views

An inequality by Hardy

Young's inequality for convolutions states that if $1 \leq p, q, r \leq \infty$ satisfy $$\frac{1}{q} + 1 = \frac{1}{p} + \frac{1}{r}$$ for all $f \in L^p(G)$ and all $g \in L^r(G)$ where $g$ and ...
1
vote
0answers
233 views

Link between time and complex amplitude argument, DFT

I have some discrete points, with time $t_n$ and value $u(t_n)$. I perform discrete fourier transormation using values $u(t_n)$. Now I have some complex values, as I understood absolute value is ...
1
vote
3answers
796 views

Is a function analytical on C iff its Fourier-transform vanishes for negative frequencies?

I think Cauchy's integral formula and the Hilbert transform can be used to prove one direction, but is this an equivalence or only an implication? edit for clarification: Is a function $f : \mathbb C ...
2
votes
1answer
755 views

Recovering the probability mass function from the characteristic function of a discrete probability distribution using Mathematica

I would like to recover the probability mass function (pmf) from the characteristic function (CF) of a discrete probability distribution using Mathematica. Ideally, I'd like to do calculations like ...
5
votes
1answer
436 views

Spectrum of a convolution operator

Let $T$ be the operator from $L^2(\mathbb R^n)$ to $L^2(\mathbb R^n)$ that is given by $Tf := f * g$ where $g$ is in $L^2$. How do I now find that the spectrum of $T$ is equal to the essential range ...
2
votes
1answer
510 views

Fixed point Fourier transform (and similar transforms)

The Fourier transform can be defined on $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$, and we can extend this to $X:=L^2(\mathbb{R}^n)$ by a density argument. Now, by Plancherel we know that ...
5
votes
3answers
1k views

Proving commutativity of convolution $(f \ast g)(x) = (g \ast f)(x)$

From any textbook on fourier analysis: "It is easily shown that for $f$ and $g$, both $2 \pi$-periodic functions on $[-\pi,\pi]$, we have $$(f \ast g)(x) = \int_{-\pi}^{\pi}f(x-y)g(y)\;dy = ...
6
votes
6answers
1k views

Fourier Analysis textbook recommendation

I am taking a fourier analysis course at the graduate level and I am unhappy with the textbook (Stein and Shakarchi). What I am looking for is a book that is less conversational and more to the ...
2
votes
1answer
660 views

Trying to derive two dimensional version of Parseval's theorem (for real valued functions)

I'm trying to express the integral $$I = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} f(x_1, x_2) \; g(x_1, x_2) \; \mathrm{d}x_1 \mathrm{d}x_2$$ of two real valued functions $f(x_1,x_2)$ and ...
1
vote
4answers
1k views

Find the Frequency Components of a Time Series Graph

For a periodic ( and not so periodic) function, it is always possible to use fourier series to find out the frequencies contained in the function. But what about function that cannot be expressed in ...
3
votes
3answers
722 views

Hint on how to prove $\zeta ( 2) =\pi ^{2}/6$ using the complex Fourier series of $f(x)=x$

I know how to prove $\zeta (2)=\pi ^{2}/6$ by using the trigonometric Fourier series expansion of $x^{2}/4$. How can one prove the same result using the complex Fourier series of $f(x)=x$ for $0\leq ...
4
votes
1answer
1k views

Wiener filter: A good tutorial

I am interested in image analysis and am looking for an approachable tutorial to the Wiener filter. At some point I am interested in implementing such a filter but I would like to have a deeper ...